You don't need all the islands! You want a naval base that is either:
A) In a good position to exert direct pressure on your enemy's fleet and naval bases.
B) Is controlled by the enemy and is being used to exert direct pressure on your fleet.
Taking the islands is generally the easier part, but its what you do afterwards that is actually useful. Namely you need Naval or Torpedo bombers there and you must use it as a base to sink as many enemy ships as possible. Its forward position will help you save your own ships & subs from complete destruction while you are using them to blockade enemy islands. Basically you surround an enemy island with ships so that they destroy his convoys, eating up all its incoming oil and supplies. This makes the garrison weaken and eventually die if you've got the patience or don't have the marines to finish it off. Even if you are attacked by enemy ships who retreat to the cover of the harbour, maintaining your blockade means that those ships will not heal or regain org, which means you have written them off, but only as long as you can keep the island blockaded. Throughout all of this your blockade ships are going to get worn down, which is why you want a friendly forward naval base to retire to, and to send reinforcements from.
It's a whole lot of back and forth action between your bases and the enemy's bases. If you are doing it effectively, the blockaded island tends to become a magnet for enemy warships which is exactly what you want, assuming you have the resources to maintain your presence there. And there will be enemy super stacks coming your way from time to time, so like I said, you will need to keep rotating battle fleets in and out of the contested waters so they can regain their org at the very least, and preferably have some of the holes in their hulls plugged.
The key here is that unless you have the air and naval resources to play at this game of attrition you are better off NOT going after the next island. It will simply spread you out further, dilute your forces, and leave them open to chance encounters with AI fleets. Chance encounters are your worst nightmare, since with no stacking penalty the AI has a tendency to win them, but even if you do win the initial battle, you won't have the Naval/Torpedo bombers nearby to exploit your success. There is no point engaging your surface fleets unless you have the airpower to finish off the enemy ships afterwards. I really mean that, it is a waste of shipping and a waste of time. If you're playing Japan you do not have enough resources to allow this type of random encounter to happen very often, and certainly the loss of a few carriers generally spells the eventual defeat of Japan. If you are going to fight a fleet action, you MUST turn it into a complete defeat of the enemy shipping involved. Using certain key islands as naval bases allows you to set up everything just so, and for a country like Japan, as is historical, it is the only way Japan can win strategically in a war with the US.